Problem: Factor the following expression: $5$ $x^2$ $-3$ $x$ $-8$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(-8)} &=& -40 \\ {a} + {b} &=& & & {-3} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-40$ and add them together. Remember, since $-40$ is negative, one of the factors must be negative. The factors that add up to ${-3}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-8}$ and ${b}$ is ${5}$ $ \begin{eqnarray} {ab} &=& ({-8})({5}) &=& -40 \\ {a} + {b} &=& {-8} + {5} &=& -3 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-8}x +{5}x {-8} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-8}x) + ({5}x {-8}) $ Factor out the common factors: $ x(5x - 8) + 1(5x - 8) $ Notice how $(5x - 8)$ has become a common factor. Factor this out to find the answer. $(5x - 8)(x + 1)$